Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13618 |
290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,61,64,79)(59,66,70,69)(60,72,75,76)(62,77,83,113)(63,78,86,106)(65,87,90,89)(67,91,95,94)(68,92,97,84)(71,99,101,100)(73,102,104,108)(74,85,107,109)(80,110,98,114)(81,115,116,111)(82,112,103,121)(88,117,118,96)(93,119,120,105);(58,60,64,75)(59,65,70,90)(61,76,79,72)(62,81,83,116)(63,85,86,109)(66,89,69,87)(67,93,95,120)(68,96,97,117)(71,98,101,80)(73,103,104,82)(74,106,107,78)(77,111,113,115)(84,88,92,118)(91,105,94,119)(99,110,100,114)(102,112,108,121);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,56,57)(58,59)(60,71)(61,66)(62,80)(63,84)(64,70)(65,73)(67,82)(68,78)(69,79)(72,99)(74,105)(75,101)(76,100)(77,110)(81,96)(83,98)(85,93)(86,92)(87,102)(88,115)(89,108)(90,104)(91,112)(94,121)(95,103)(97,106)(107,119)(109,120)(111,118)(113,114)(116,117)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=df,f^b=fg,g^b=gh1196 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90015);